The isomorphism problem for cyclic algebras and an application

ABSTRACT

The isomorphism problem means to decide if two given finite-dimensional simple algebras with center K are K-isomorphic and, if so, to construct a K-isomorphism between them. Applications lie in computational aspects of representation theory, algebraic geometry and Brauer group theory. The paper presents an algorithm for cyclic algebras that reduces the isomorphism problem to field theory and thus provides a solution if certain field theoretic problems including norm equations can be solved (this is satisfied over number fields). As an application, we can compute all automorphisms of any given cyclic algebra over a number field. A detailed example is provided which leads to the construction of an explicit noncrossed product division algebra.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

General Terms:
Design, Theory

Keywords:
abelian crossed product, bicyclic crossed product, cyclic algebra, extension of automorphism, finite-dimensional central-simple algebra, isomorphism problem, noncrossed product, norm equation